Certainly! Let’s define even and odd numbers and outline their properties:
Even Number: An even number is an integer that is exactly divisible by 2 without leaving a remainder. In other words, if a number can be expressed as (2 \times k), where (k) is an integer, then it is an even number.
Odd Number: An odd number is an integer that is not exactly divisible by 2, meaning it leaves a remainder of 1 when divided by 2. In other words, if a number can be expressed as (2 \times k + 1), where (k) is an integer, then it is an odd number.
Properties of Even Numbers:
- Divisibility by 2: Every even number is divisible by 2.
- Addition: The sum of two even numbers is always even.
- ( \text{Even} + \text{Even} = \text{Even} )
- Subtraction: The difference between two even numbers is always even.
- ( \text{Even} - \text{Even} = \text{Even} )
- Multiplication: The product of two even numbers is always even.
- ( \text{Even} \times \text{Even} = \text{Even} )
- Division: When an even number is divided by 2, the result is always an integer.
- ( \frac{\text{Even}}{2} = \text{Integer} )
Properties of Odd Numbers:
- Divisibility by 2: An odd number is not divisible by 2.
- Addition: The sum of two odd numbers is always even.
- ( \text{Odd} + \text{Odd} = \text{Even} )
- Subtraction: The difference between two odd numbers may be even or odd.
- ( \text{Odd} - \text{Odd} = \text{Even} ) or ( \text{Odd} - \text{Odd} = \text{Odd} )
- Multiplication: The product of two odd numbers is always odd.
- ( \text{Odd} \times \text{Odd} = \text{Odd} )
- Division: When an odd number is divided by 2, the result is not necessarily an integer.
- ( \frac{\text{Odd}}{2} = \text{Not an Integer} )
Perfect Number:
- A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding itself).
- Mathematically, a perfect number (N) satisfies the equation: (\sigma(N) = 2N), where (\sigma(N)) represents the sum of divisors of (N).
- Example: The first few perfect numbers are 6, 28, 496, and 8128.
Prime Number:
- A prime number is a positive integer greater than 1 that has exactly two distinct positive divisors: 1 and itself.
- Example: 2, 3, 5, 7, 11, 13, etc., are prime numbers.
Twin Prime:
- Twin primes are pairs of prime numbers that have a difference of 2.
- Examples: (3, 5), (5, 7), (11, 13), (17, 19), etc., are twin prime pairs.
Relative Prime (Coprime):
- Two integers are said to be relatively prime or coprime if they have no common positive integer divisors other than 1.
- Example: (4, 9) are coprime because their only common divisor is 1. However, (6, 8) are not coprime because they have a common divisor of 2.
Composite Number:
- A composite number is a positive integer greater than 1 that is not prime, meaning it has more than two distinct positive divisors.
- Example: 4, 6, 8, 9, 10, 12, etc., are composite numbers.
Divisibility
Divisibility by 2:
- A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
Divisibility by 3:
- A number is divisible by 3 if the sum of its digits is divisible by 3.
Divisibility by 4:
- A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
Divisibility by 5:
- A number is divisible by 5 if its last digit is either 0 or 5.
Divisibility by 6:
- A number is divisible by 6 if it is divisible by both 2 and 3.
Divisibility by 7:
- There isn’t a straightforward rule like other numbers, but methods like “double the last digit and subtract” can be used for divisibility by 7.
Divisibility by 8:
- A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
Divisibility by 9:
- A number is divisible by 9 if the sum of its digits is divisible by 9.
Divisibility by 10:
- A number is divisible by 10 if its last digit is 0.
Divisibility by 11:
- A number is divisible by 11 if the alternating sum of its digits is divisible by 11. (The alternating sum is obtained by subtracting the second digit from the first, adding the third digit, subtracting the fourth, and so on.)